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## Homework Statement

Prove that if ##H\leq K\leq G## and ##[G : K]## and ##[K : H]## are finite, then ##[G:K]\cdot [K : H]=[G : H]##

## Homework Equations

## The Attempt at a Solution

Here is my attempt. We first note that $$G = \bigcup_{x\in G} xK ~\text{and}~ K=\bigcup_{y\in K}yH$$.

So $$G = \bigcup_{x\in G} x\bigcup_{y\in K}yH = \bigcup_{(x,y)\in G \times K}xyH$$ Now we show that ##G## is a disjoint union. Suppose that ##xyH \cap x'y'H \not = \emptyset##. Then there is some nontrivial ##x## such that ##x = xyh=x'y'h'## for some ##h,h'\in H##. But then it's clear that ##xyH=x'y'H##. Hence we have a disjoint union.

This is where I get stuck. Is there some way I can glean from what I proved the fact that ##[G:K]\cdot [K : H]=[G : H]##?